**SHARPEST TRIANGLES**

Excel file with Horizontal bases: 7.H_SharpestRightTrianglesOnEvenPolygons

Excel file with *Slanted* bases: 7.S_SharpestRightTrianglesOnEvenPolygons

Partial Paper: SharpestAngleRightTriangles

Challenge Question: CQ Sharpest Right Triangles with 4 sets of lines

The files here are modified versions of File 6 which looked at sharpest isosceles triangles on odd-polygons. This is a natural extension of the material discussed there and formulas developed there are quickly used to answer how many sharpest angle right triangles occur when ** n** is even. Interestingly, two distinct images emerge for each value of

**, one has horizontal bases with slanted hypotenuse and the other with slanted base and vertical hypotenuse. Neither the files or the paper above are in polished shape but they do show the images and provide a quick synopsis of what is going on here.**

*n*The explainers below are more polished and they go beyond the sharpest right triangles on even polygons case which starts off this chapter. The rest of the chapter uses the general *Excel* file from File 9 to create images rather than relying on individual files like was done for isosceles triangles on odd polygons or right triangles on even polygons.

**P4. Sharpest Triangles**

Right Triangles on Even Polygons

- Sharpest Right Triangles on Even Polygons
- How Many Triangles are in the Sharpest Right Triangles Images
- This handout works for sharpest right triangles: 7.Counting Handout

Obtuse Triangles

- What we mean by Largest Large Triangles and Smallest Large Triangles
- Any Sharpest Triangle that is not Isosceles or Right is Obtuse
- Sharpest Most Obtuse Isosceles and Scalene Triangles Images
- Sharpest Least Obtuse Scalene Triangles Images
- General Sharpest Triangles as Deviations from Sharpest Isosceles or Right Triangles
- Counting General Sharpest Triangles by Focusing on Angles